# kernel

Let $\mathrm{\Sigma}$ be a fixed signature^{}, and $\U0001d504$ and $\U0001d505$ be two structures^{} for $\mathrm{\Sigma}$. Given a homomorphism^{} $f:\U0001d504\to \U0001d505$, the *kernel* of $f$ is the relation^{} $\mathrm{ker}(f)$ on $A$ defined by

$$\u27e8a,{a}^{\prime}\u27e9\in \mathrm{ker}(f)\iff f(a)=f({a}^{\prime}).$$ |

So defined, the kernel of $f$ is a congruence^{} on $\U0001d504$. If $\mathrm{\Sigma}$ has a constant symbol 0, then the kernel of $f$ is often defined to be the preimage^{} of ${0}^{\U0001d505}$ under $f$. Under this definition, if $\{{0}^{\U0001d505}\}$ is a substructure of $\U0001d505$, then the kernel of $f$ is a substructure of $\U0001d504$.

Title | kernel |
---|---|

Canonical name | Kernel1 |

Date of creation | 2013-03-22 13:46:34 |

Last modified on | 2013-03-22 13:46:34 |

Owner | almann (2526) |

Last modified by | almann (2526) |

Numerical id | 11 |

Author | almann (2526) |

Entry type | Definition |

Classification | msc 03C05 |

Classification | msc 03C07 |

Related topic | Kernel |

Related topic | KernelOfAGroupHomomorphism |

Related topic | KernelOfALinearTransformation |